可积模的权

本文定义了Ｋａｃ－Ｍｏｏｄｙ代数的一个新的可积模范畴，并且给出了一个可积模是否属于这个模范畴的一个判别准则．另外还详细研究了这个模范畴中的可积模的权系。特别我们定义了虚权和实权。还详细地计算了一些模的虚权和实权，还给出了双曲型广义Ｃａｒｔａｎ矩阵的新刻划．这使我们能够计算一些双曲型Ｋａｃ－Ｍｏｏｄｙ代数的可积模的权。     

绝对Henstock可积函数都是Mcshane可积的

我们给出每个绝对Ｈｅｎｓｔｏｃｋ可积函数都是Ｍｃｓｈａｎｅ可积的一个新的证明     

Explicit Solutions to an Integrable Lattice

In this paper, we aim to construct the Darboux transformation and explicit solutions for an integrable lattice introduced by Suris. Analysis of properties of the solutions shows that the obtained explicit solutions for this discrete integrable system possess new dynamical characters which are different from the ones of continuous integrable systems.     

一个类似于KN族的可积系及其可积耦合

本文选用loop代数A₁的一个子代数,建立了一个线笥等谱问题,导出了一个类似KN族的可积方程族．通过建立求可积耦合的一种简便直接方法,求出了该方程族的可积耦合．这种方法也适用于其它方程族。     

Integrable elliptic pseudopotentials

We construct integrable pseudopotentials with an arbitrary number of fields in terms of an elliptic generalization of hypergeometric functions in several variables. These pseudopotentials are multiparameter deformations of ones constructed by Krichever in studying the Whitham-averaged solutions of the KP equation and yield new integrable (2+1)-dimensional systems of hydrodynamic type. Moreover, an interesting class of integrable (1+1)-dimensional systems described in terms of solutions of an elliptic generalization of the Gibbons-Tsarev system is related to these pseudopotentials.     

多分量AKNS方程的新可积分解

从一个任意阶矩阵谱问题出发,多分量AKNS方程的新可积分解被导出.通过利用迹恒等式建立了其双哈密顿结构.同时,证明了空间与时间的约束流在刘维尔意义下是两个完全可积的哈密顿系统.     

Integrable variational equations of non-integrable systems

Paper is devoted to the solvability analysis of variational equations obtained by linearization of the Euler-Poisson equations for the symmetric rigid body with a fixed point on the equatorial plain. In this case Euler-Poisson equations have two pendulum like particular solutions. Symmetric heavy top is integrable only in four famous cases. In this paper is shown that a family of cases can be distinguished such that Euler-Poisson equations are not integrable but variational equations along particular solutions are solvable. The connection of this result with analysis made in XIX century by R. Liouville is also discussed.     

Integrable Geodesic Flows on Surfaces

We propose a new condition à{{\aleph}} which enables us to get new results on integrable geodesic flows on closed surfaces. This paper has two parts. In the first, we strengthen Kozlov’s theorem on non-integrability on surfaces of higher genus. In the second, we study integrable geodesic flows on a 2-torus. Our main result for a 2-torus describes the phase portraits of integrable flows. We prove that they are essentially standard outside what we call separatrix chains. The complement to the union of the separatrix chains is C ⁰-foliated by invariant sections of the bundle.     

Integrable pseudopotentials related to generalized hypergeometric functions

We construct integrable pseudopotentials with an arbitrary number of fields in terms of generalized hypergeometric functions. These pseudopotentials yield some integrable (2 + 1)-dimensional hydrodynamic type systems. In two particular cases these systems are equivalent to integrable scalar 3-dimensional equations of second order. An interesting class of integrable (1 + 1)-dimensional hydrodynamic type systems is also generated by our pseudopotentials.     

利用李代数构造非线性可积及双可积哈密顿耦合

With the help of a Lie algebra,two kinds of Lie algebras with the forms of blocks are introduced for generating nonlinear integrable and bi-integrable couplings.For illustrating the application of the Lie algebras,an integrable Hamiltonian system is obtained,from which some reduced evolution equations are presented.Finally,Hamiltonian structures of nonlinear integrable and bi-integrable couplings of the integrable Hamiltonian system are furnished by applying the variational identity.The approach presented in the paper can also provide nonlinear integrable and bi-integrable couplings of other integrable system.     

Integrable Quasilinear Equations

We develop a classification scheme for integrable third-order scalar evolution equations using the symmetry approach to integrability. We use this scheme to study quasilinear equations of a particular type and prove that several equations that were suspected to be integrable can be reduced to the well-known Korteweg–de Vries and Krichever–Novikov equations via a Miura-type differential substitution.     

G可积函数的Lebesgue可测性

Botsko在连续和可导的知识基础上推广了Riemann积分,得到了一种新的积分,称为G积分．G积分既不同于Riemann积分也不同于Lebesgue积分．本文通过对G积分的研究,得到了G可积函数一定Lebesgue可测,从而有界G可积函数一定Lebesgue可积;同时我们还证明了这两个积分值相等．     

On Integrable roots in split Lie triple systems

We focus on the notion of an integrable root in the framework of split Lie triple systems T with a coherent 0-root space. As a main result, it is shown that if T has all its nonzero roots integrable, then its standard embedding is a split Lie algebra having all its nonzero roots integrable. As a consequence, a local finiteness theorem for split Lie triple systems, saying that whenever all nonzero roots of T are integrable then T is locally finite, is stated. Finally, a classification theorem for split simple Lie triple systems having all its nonzero roots integrable is given.     

一族新的Lax可积系及其Liouville可积性

该文讨论了一个新的等谱特征值问题．按屠规彰格式导出了相应的Lax可积的非线性发展方程族，利用迹恒等式给出了它的Hamilton结构并且证明它是Liouville可积的．     

Integrable many-body systems and gauge theories

We review the study of the relation between integrable many-body systems and gauge theories. We show that the degrees of freedom of integrable systems are related to the topological degrees of freedom of gauge theories. We also describe the relation between families of integrable systems and N=2 supersymmetric gauge theories. We show that the degrees of freedom of many-body systems can be identified with the collective coordinates of string theory solitons, theD-branes. This article was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 3–56, October, 2000.     

Integrable lattices

We propose a method for constructing integrable lattices starting from dynamic systems with two different parameterizations of the canonical variables and hence two independent Bäcklund flows. We construct integrable lattices corresponding to generalizations of the nonlinear Schrödinger equation. We discuss the Toda, Volterra, and Heisenberg models in detail. For these systems, as well as for the Landau-Lifshitz model, we obtain totally discrete Lagrangians. We also discuss the relation of these systems to the Hirota equations.     

Integrable structures of dispersionless systems and differential geometry

We develop the theory of Whitham-type hierarchies integrable by hydrodynamic reductions as a theory of certain differential-geometric objects. As an application, we construct Gibbons–Tsarev systems associated with the moduli space of algebraic curves of arbitrary genus and prove that the universal Whitham hierarchy is integrable by hydrodynamic reductions.     

Induced Surfaces and Their Integrable Dynamics

A method is considered to induce surfaces in three-dimensional (pseudo) Euclidean space via the solutions to two-dimensional linear problems (20 LPs) and their integrable dynamics (deformations) via the 2 + 1-dimensional nonlinear integrable equations associated with these 2D LPs. Coordinates X_i of the induced surfaces are defined as integrals over certain bilinear combinations of the wave functions ψ of these 20 LPs. General formulation as well as three concrete examples are considered. Some properties and features of such induction are discussed. Three-dimensional Riemann spaces associated with 2 + 1-dimensional nonlinear integrable equations are considered also.     

双约束孤立子流的可积形变

提出了基于Lax矩阵的构造双约束孤立子流的可积形变的新方法.作为应用,导出了双约束KdV流和双约束mKdV流的可积形变,并给出了这些形变的Lax表示、r-矩阵和守恒积分.